An upper bound on tricolored ordered sum-free sets

TL;DR

This paper introduces a strengthened lemma based on upper-triangular matrix rank to improve bounds on tricolored ordered sum-free sets in finite fields, extending previous cap set results.

Contribution

The paper develops a new lemma on matrix rank that enhances the bound on sum-free sets, extending Ellenberg-Gijswijt's cap set bounds to a tricolored ordered setting.

Findings

Established an upper bound on the size of tricolored ordered sum-free sets in _p^n.

Extended the Ellenberg-Gijswijt bound to a new class of sum-free sets.

Introduced a novel matrix rank approach based on upper-triangular matrices.

Abstract

We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in $\mathbb F_p^n$ is a collection $\{(a_i,b_i,c_i):i=1,2,\ldots,m\}$ of ordered triples in $(\mathbb F_p^n )^3$ such that $a_i+b_i+c_i=0$ and if $a_i+b_j+c_k=0$, then $i\le j\le k$. By using the new lemma, we present an upper bound on the size of a tricolored ordered sum-free set in $\mathbb F_p^n$.